Don't know much about suspenders and didn't know Hold Ups were a brand until Twelvepole pointed it out.
Looking at the pic on that website, I'd think the bottom of the suspender is just folded around the clip, then stitched in place.
If you don't want to send it back for a replacement, you can undo the stitching with a seam ripper, then if you have a heavy duty sewing machine with a #18 needle, you can insert the new clip, fold it up and restitch with a topstitching thread.
If you don't sew, you could take it to a shoemaker (if you can find one) who could do this.